Explicit sufficient stability conditions on dwell time of linear switched systems

Explicit conditions on the minimum dwell time that guarantees the asymptotic stability of switched linear systems are given. To this aim, methods that have been proposed for non-defective stable subsystem matrices are generalized to arbitrary stable subsystems matrices. Admissible switchings between subsystems are assumed to be in a general form, namely switchings respect a given directed graph. It is shown that logarithmic norm of matrix exponentials and Lambert-W functions can be used to bound the solutions of switched linear systems in case of defective subsystem matrices. Using a generalized version of Jordan form, dwell time bound can be found for any set of stable subsystem matrices.